Answer :
the answer is (A) 12.8 minutes.
We can use Newton's Law of Cooling to solve this problem. This law states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings. In this case, the object is the hot tea, and its surroundings are the room temperature.
Let t be the time (in minutes) it takes for the tea to cool from 212°F to 160°F. Then we have:
Rate of cooling = k(Tea temperature - Room temperature)
where k is the proportionality constant and is given as 0.05 in the problem.
At the start, the tea temperature is 212°F, so we have:
0.05(212 - 68) = dT/dt
Simplifying this, we get:
dT/dt = 7.2
This means that the tea is cooling at a rate of 7.2°F per minute at the start.
Now, we want to find how long it takes for the tea to cool from 212°F to 160°F. Let's integrate the above equation between the initial temperature of 212°F and the final temperature of 160°F, and the initial time of 0 minutes and the final time t:
∫(212-68)/(160-68) dT = ∫0²t 0.05 dt
Simplifying this, we get:
2.5 ln(3/4) = 0.05t
Solving for t, we get:
t = (2.5 ln(3/4))/0.05 ≈ 12.8 minutes
Therefore, the answer is (A) 12.8 minutes.
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